Good Banach spaces for piecewise hyperbolic maps via interpolation

By Viviane Baladi, S. Gouëzel. 2007

70 citations recorded by the source corpus.

We introduce a weak transversality condition for piecewise C 1+α and piecewise hyperbolic maps which admit a C 1+α stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces. © 2009 Elsevier Masson SAS. All rights reserved.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a400237749747c96a8f2f7e08383405349d2e1a7

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In Mathematical Physics

1. C. Gonz'alez-Tokman
2. Anthony Quas
3. A. Blumenthal
4. Gary Froyland

In Statistical and Nonlinear Physics

1. C. Gonz'alez-Tokman
2. Anthony Quas
3. Gary Froyland

Citations in the corpus

• A semi-invertible operator Oseledets theorem (2011, methodology): “The necessity of controlling Cb is demonstrated by results of Tsujii [45] and Buzzi [8], and indeed Cb appears in Baladi and Gouëzel [2].”
• Banach spaces for piecewise cone hyperbolic maps (2009, unknown): “These results are mostly adapted from [BG09].”